[Optimality is inherited by restriction [1]}]

Let \((X,\mu )\)  and \((Y,\nu )\)  be two Polish spaces; let \(a:X\mapsto \mathbb {R}~\cup \lbrace -\infty \rbrace \)  and \(b:Y\mapsto \mathbb {R}~\cup \lbrace -\infty \rbrace \)  be two upper semi-continuous functions such that \(a\in L^1(\mu )\) , \(b\in L^1(\nu )\) .
Let \(c:X\times Y\mapsto \mathbb {R}~\cup \lbrace +\infty \rbrace \)  be a lower semi-continuous cost function, such that \(c(x,y)\ge a(x)+b(y)\)  for all \(x,y\) .
Let \(C(\mu ,\nu )\)  be the optimal transport cost from \(\mu \)  to \(\nu \) . Assume that \(C(\mu ,\nu )<\infty \)  and let \(\gamma \in \Pi (\mu ,\nu )\)  be an optimal coupling.
Let \(\tilde{\gamma }\)  be a non-negative measure on \(X\times Y\)  such that \(\tilde{\gamma }\le \gamma \)  and \(\tilde{\gamma }[X\times Y]>0\) .
Then the probability measure
\(\gamma ^{\prime }:= \frac{\tilde{\gamma }}{\tilde{\gamma }[X\times Y]}\) 
